QUESTION:

 

Please explain the notion of TRANSFER FUNCTION identification 

 

and the role of the pre-whitening filter as a tool in this step ? 

 

ANSWER:

 

Consider a Transfer Function of the form: 

 

Y(t ) = V0 X(t ) + V1 X(t-1) + V2 X(t-2) .... + Vj X(t-j) + N(t ) 

 

or Y(t ) = [W(B)/D(B)] X(t ) + [T(B)/P(B)] A(t ) 

 

where an ARIMA model for X exists such that 

 

E(t ) = [T1(B)/P1(B)] X(t ) 

 

equivalently 

 

X(t ) = [P1(B)/T1(B)] E(t ) 

 

Now more generally our Transfer Function may contain a pure delay of b periods 

 

Y(t ) = [W(B)/D(B)] X(t-b) + [T(B)/P(B)] A(t ) 

 

where 

 

b = the number of periods of pure delay before Y responds to X 

 

{T(B)/P(B)} = ARMA model for unobserved series A 

 

W(b) = input lag structure reflecting static relationship of 

 

Y to X and is a polynomial of order s-1 

 

D(b) = output lag structure reflecting dynamic relationship of 

 

Y to X and is a polynomial of order r 

 

and substituting the stationary y and Y for x and X we get: 

 

y(t) = [W(B)/D(B)] x(t-b) + [T(B)/P(B)] A(t) [1] 

 

If we now identify the following ARMA for x 

 

E(t) = [T1(B)/P1(B)] x(t) [2] 

 

If we now multiply the equation [1] above by [T1(B)/P1(B)] 

 

[T1(B)/P1(B)]y(t ) = [W(B)/D(B)][T1(B)/P1(B)] x(t-b) + [T1(B)/P1(B)][T(B)/P(B)] A(t ) 

 

Substituting [2] into the current equation generates 

 

[T1(B)/P1(B)]y(t ) = [W(B)/D(B)] E(t-b) + [T1(B)/P1(B)][T(B)/P(B)] A(t) 

 

If we let 

 

y(t )=[T1(B)/P1(B)]y(t ) and aa(t )=[T1(B)/P1(B)][T(B)/P(B)] A(t ) 

 

we have 

 

yy(t ) = [W(B)/D(B)] E(t-b) + aa(t ) 

 

where E(t ) is N.I.I.D. Thus we get a clear picture 

 

of [W(B)/D(B)] which represents the Transfer of X into Y. 

 

In practice we identify [W(B)/D(B)] via the cross-correlations of 

 

yy(t ) and E(t ) , selecting orders of [W(B)/D(B)] sufficient to 

 

describe the relationship. The pure delay (b) represents the observed 

 

dead time. Upon identifying [W(B)/D(B)] we can then back out the effect 

 

of X on Y and obtain an initial estimate of N(t ) from: 

 

Y(t ) = V0 X(t ) + V1 X(t-1) + V2 X(t-2) .... + Vj X(t-j) + N(t ) 

 

Y(t) - V0 X(t) - V1 X(t-1) - V2 X(t-2) .... - Vj X(t-j) = N(t ) 

 

We can now study N(t) and obtain the two polynomials T(B) and P(B). 

 

N(t) = [T(B)/P(B)] A(t) which completes the identification step. 

 

This explains the need for filtering and exposes the relationship 

 

of the prewhitening filter in identifying the form of the inter-relationship 

 

between the observed series Y and X. 

 

Y(t) = [W(B)/D(B)] X(t) + [T(B)/P(B)] A(t)